sathwik_nallamada_final_project
sathwik_reddy
Introduction
1. What is the source of your dataset?
The dataset I considered is taken from the UCI machine learning repository.
2. What is the context in which it was collected?
The dataset was collected using analyzing the different case studies and categorizing the diseases according to the nature of their intensity. The dataset is collected from 270 participants of different age groups, genders, etc. by the use of health tracking devices connected via IoT (Internet of things)
In this dataset, we have different predictor variables for scrutinizing and predicting the different heart-related diseases. These are measured across the 11 different parameters. That includes age, sex, chest pain type,BP, cholesterol,and FBS.over 120,EKG results, Max.HR, exercise. Enigma, ST depression, number of vessels Fluro, thallium, and heart disease.
3. Do you have permission to share the dataset and its analyses?
Since this an opensource dataset I have the permission to share and analyse the dataset.
4. What approach was used to compile this dataset?
The approach I followed is the linear regression approach where I considered the different number of predictor variables in 4 different models. The first model I considered is a full model where it has all the necessary values and omits the rest. The second model that I considered is the predictor variables that I am most interested in. The third model contains the filtered predictor variables from the 2nd model that has a significant relation with the heartbeat.The fourth model contains the variables among the three models that gave a significant relationship to the human heart rate (Max. HR).
5. Present details of the variables in your dataset (include both the original variables and any new variables you create via transformation and/or combination of the original dataset. Include details of the transformation and computations in the description column.
| SNo | Name | Description | measurement type | Role |
|---|---|---|---|---|
| 1 | Age | Age of the participant(in years) | Numeric | Predictor |
| 2 | Sex | Gender of the participant | Categorical | Predictor |
| 3 | Blood pressure | The value of the resting blood pressure. | Numeric | Predictor |
| 4 | Cholestrol | The serum cholesterol level in the body .Measured by mg/dl | Numeric | Predictor |
| 5 | Chest pain type | 1-typical angina 2-Atypical angina 3-non-anginal pain 4-asymptotic |
Categorical | Outcome |
| 6 | FBS over 120 | Blood sugar level | numeric | Outcome |
| 7 | ST depression | The level of internal depression of the participant | Numeric | Outcome |
| 8 | Number of vessels | The number of vessels in the heart narrowing down | Numeric | Outcome |
| 9 | The allium | The allium heart related disease categorized by 3-normal 4-fined effect 7-reversible efffect |
Numeric | Outcome |
| 10 | Max.HR | The maximum human heart rate | Numeric | Predictor |
| 11 | Heart disease | The presence or absence of heart disease | Categorical | Outcome |
- Provide a list of two-three research questions. These should require non-summary type of analyses, via building appropriate models and interpreting their results, to be answered.
The problem statement that I am going to mainly address is ’How the data collected by IoMT devices specifically heart rate can be used to predict the diseases in the human body before they mature, and cause severe illnesses,
The questions associated with this problem statement are :
How accurate is the prediction of illness in the human body by measuring the fluctuations in the human heart rate?
What is the greatest number of disparities in the human body that can be predicted by analyzing the human heart rate?
Conceptual model
- Begin by providing a data model, indicating, on the basis of the nature of the outcome variable, the specific type of GLM you intend to use.
The models that are going to scrutinize are four different models one all of the predictors included, two my preferred predictor variables, three the most accurate predictors model 3, and the most significant predictors among the three models.
The reason behind choosing and structuring this model as the way mentioned above is to know the most significant predictors in the first place that can be predicted using the predictor variable Max. HR.
The second model is the ideal model that I want to examine. It contains the predictors where the severity of illness is high compared with the rest.
The third model contains filters out the predictor variables from the second model, which has a significant correlation with the human heart rate.
The fourth model contains the amalgamation of the most significant predictor variables from the above models and the filtered variables of my preference from the second model.
The purpose behind designing this model in this order is to gain the correct balance between the preferred choice of variables and the computer-generated results.
The specific GLM I am looking forward to use is
Y|β0,β1,β2 with the standard deviation sigma.
μ=β0+β1X1+β2X2+ β3X3+β4X4
with Y=MAX.HR(maximum heart rate )
2) Provide a specification of the prior PDFs for each of the model parameters.
The prior probability density functions for all the four models are the range the of the heart rate .The range of the Max.H.R can be determined by using summary function on the dataset I selected .
Beta (Max.HR , all_predictor variables ) ∼ N(m0,s.d^2)
Beta (Max.HR,Age+BP+FBS.over.120+Cholesterol+Sex) ∼ N(m1, s.d^2)
Beta (Max.HR,FBS.over.120+Cholesterol+Sex) ∼ N(m2, s.d^2)
Beta(Max.HR,Chest.pain.type+FBS.over.120+Slope.of.ST+Number.of.vessels.fluro) ∼ N(m3, s.d^2)
3)Provide a listing of hypotheses, one for each of the model parameters, in their null form. Provide a rationale for why these specific relationships are chosen to be the null (e.g., the relationship between number of cigarettes smoked and probability of developing cancer is assumed to have a coefficient of at least 1.3. What is the basis for this specific value of the beta coefficient associated with number of cigarettes smoken? Reference a particular study or set of studies where this model was shown to be the case).
The listing hypothesis for each of the model parameters in their null form are:
H0-For my first model I am testing is, there is no relation between the age and sex parameters with the Max.HR.
HO1-For the second model I am testing is, there is no significant relation with the Blood pressure(BP) with the Max.HR.
H02For the third model I am testing id the there is no association with the cholesterol and the MAX.HR
4)Provide a tuning for the prior PDFs’ parameter values. This will require you to derive these values from prior knowledge and null hypothesis statements.
The prior values I considered are the range if the human heart rate .The tuning of the models are been done by analyzing the summary function of the first model.The first model contains all the predictor variables and the turning of the model are modified accordingly based on the summary function of the first model.
β0-N(71,65.5)
β1~N(0,1)
library(bayesrules)
library(tidyverse)## ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.1 ──## ✔ ggplot2 3.3.6 ✔ purrr 0.3.4
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## ✔ readr 2.1.2 ✔ forcats 0.5.1## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()library(bayesplot)## This is bayesplot version 1.9.0## - Online documentation and vignettes at mc-stan.org/bayesplot## - bayesplot theme set to bayesplot::theme_default()## * Does _not_ affect other ggplot2 plots## * See ?bayesplot_theme_set for details on theme settinglibrary(rstanarm)## Loading required package: Rcpp## This is rstanarm version 2.21.3## - See https://mc-stan.org/rstanarm/articles/priors for changes to default priors!## - Default priors may change, so it's safest to specify priors, even if equivalent to the defaults.## - For execution on a local, multicore CPU with excess RAM we recommend calling## options(mc.cores = parallel::detectCores())library(broom.mixed)
library(tidybayes)
library(GGally)## Registered S3 method overwritten by 'GGally':
## method from
## +.gg ggplot2dataset <- read.csv("~/Downloads/Heart_Disease_Prediction.csv", header=TRUE)Loaded the required data set by specifying the exact path required and the header is set as true.
view(dataset)
summary(dataset)## Age Sex Chest.pain.type BP
## Min. :29.00 Min. :0.0000 Min. :1.000 Min. : 94.0
## 1st Qu.:48.00 1st Qu.:0.0000 1st Qu.:3.000 1st Qu.:120.0
## Median :55.00 Median :1.0000 Median :3.000 Median :130.0
## Mean :54.43 Mean :0.6778 Mean :3.174 Mean :131.3
## 3rd Qu.:61.00 3rd Qu.:1.0000 3rd Qu.:4.000 3rd Qu.:140.0
## Max. :77.00 Max. :1.0000 Max. :4.000 Max. :200.0
## Cholesterol FBS.over.120 EKG.results Max.HR
## Min. :126.0 Min. :0.0000 Min. :0.000 Min. : 71.0
## 1st Qu.:213.0 1st Qu.:0.0000 1st Qu.:0.000 1st Qu.:133.0
## Median :245.0 Median :0.0000 Median :2.000 Median :153.5
## Mean :249.7 Mean :0.1481 Mean :1.022 Mean :149.7
## 3rd Qu.:280.0 3rd Qu.:0.0000 3rd Qu.:2.000 3rd Qu.:166.0
## Max. :564.0 Max. :1.0000 Max. :2.000 Max. :202.0
## Exercise.angina ST.depression Slope.of.ST Number.of.vessels.fluro
## Min. :0.0000 Min. :0.00 Min. :1.000 Min. :0.0000
## 1st Qu.:0.0000 1st Qu.:0.00 1st Qu.:1.000 1st Qu.:0.0000
## Median :0.0000 Median :0.80 Median :2.000 Median :0.0000
## Mean :0.3296 Mean :1.05 Mean :1.585 Mean :0.6704
## 3rd Qu.:1.0000 3rd Qu.:1.60 3rd Qu.:2.000 3rd Qu.:1.0000
## Max. :1.0000 Max. :6.20 Max. :3.000 Max. :3.0000
## Thallium Heart.Disease
## Min. :3.000 Length:270
## 1st Qu.:3.000 Class :character
## Median :3.000 Mode :character
## Mean :4.696
## 3rd Qu.:7.000
## Max. :7.000str(dataset)## 'data.frame': 270 obs. of 14 variables:
## $ Age : int 70 67 57 64 74 65 56 59 60 63 ...
## $ Sex : int 1 0 1 1 0 1 1 1 1 0 ...
## $ Chest.pain.type : int 4 3 2 4 2 4 3 4 4 4 ...
## $ BP : int 130 115 124 128 120 120 130 110 140 150 ...
## $ Cholesterol : int 322 564 261 263 269 177 256 239 293 407 ...
## $ FBS.over.120 : int 0 0 0 0 0 0 1 0 0 0 ...
## $ EKG.results : int 2 2 0 0 2 0 2 2 2 2 ...
## $ Max.HR : int 109 160 141 105 121 140 142 142 170 154 ...
## $ Exercise.angina : int 0 0 0 1 1 0 1 1 0 0 ...
## $ ST.depression : num 2.4 1.6 0.3 0.2 0.2 0.4 0.6 1.2 1.2 4 ...
## $ Slope.of.ST : int 2 2 1 2 1 1 2 2 2 2 ...
## $ Number.of.vessels.fluro: int 3 0 0 1 1 0 1 1 2 3 ...
## $ Thallium : int 3 7 7 7 3 7 6 7 7 7 ...
## $ Heart.Disease : chr "Presence" "Absence" "Presence" "Absence" ...1)The view function is used to view the contents in the file dataset
2)The summary function is used to summarize the contents in the dataset file.
3)Str function specifies the string name from each and every variables.
dataset1=dataset%>%select(Age,BP,FBS.over.120,Cholesterol,Chest.pain.type,ST.depression,Number.of.vessels.fluro,Thallium,Sex,Max.HR)Select function is used to select the required columns from the dataset .In the above function the selected rows are transferred to the new file called dataset1
ggpairs(dataset1)
GGpairs helps to find the correlation between the contents present in the dataset1.In the above graph Max.HR is negatively correlated with most of the predictor variables while with the exception with the blood sugar level.
dataset1 %>% na.omit(dataset1)summary(dataset1)## Age BP FBS.over.120 Cholesterol
## Min. :29.00 Min. : 94.0 Min. :0.0000 Min. :126.0
## 1st Qu.:48.00 1st Qu.:120.0 1st Qu.:0.0000 1st Qu.:213.0
## Median :55.00 Median :130.0 Median :0.0000 Median :245.0
## Mean :54.43 Mean :131.3 Mean :0.1481 Mean :249.7
## 3rd Qu.:61.00 3rd Qu.:140.0 3rd Qu.:0.0000 3rd Qu.:280.0
## Max. :77.00 Max. :200.0 Max. :1.0000 Max. :564.0
## Chest.pain.type ST.depression Number.of.vessels.fluro Thallium
## Min. :1.000 Min. :0.00 Min. :0.0000 Min. :3.000
## 1st Qu.:3.000 1st Qu.:0.00 1st Qu.:0.0000 1st Qu.:3.000
## Median :3.000 Median :0.80 Median :0.0000 Median :3.000
## Mean :3.174 Mean :1.05 Mean :0.6704 Mean :4.696
## 3rd Qu.:4.000 3rd Qu.:1.60 3rd Qu.:1.0000 3rd Qu.:7.000
## Max. :4.000 Max. :6.20 Max. :3.0000 Max. :7.000
## Sex Max.HR
## Min. :0.0000 Min. : 71.0
## 1st Qu.:0.0000 1st Qu.:133.0
## Median :1.0000 Median :153.5
## Mean :0.6778 Mean :149.7
## 3rd Qu.:1.0000 3rd Qu.:166.0
## Max. :1.0000 Max. :202.0dataset1 %>% na.omit(dataset1)summary(dataset1)## Age BP FBS.over.120 Cholesterol
## Min. :29.00 Min. : 94.0 Min. :0.0000 Min. :126.0
## 1st Qu.:48.00 1st Qu.:120.0 1st Qu.:0.0000 1st Qu.:213.0
## Median :55.00 Median :130.0 Median :0.0000 Median :245.0
## Mean :54.43 Mean :131.3 Mean :0.1481 Mean :249.7
## 3rd Qu.:61.00 3rd Qu.:140.0 3rd Qu.:0.0000 3rd Qu.:280.0
## Max. :77.00 Max. :200.0 Max. :1.0000 Max. :564.0
## Chest.pain.type ST.depression Number.of.vessels.fluro Thallium
## Min. :1.000 Min. :0.00 Min. :0.0000 Min. :3.000
## 1st Qu.:3.000 1st Qu.:0.00 1st Qu.:0.0000 1st Qu.:3.000
## Median :3.000 Median :0.80 Median :0.0000 Median :3.000
## Mean :3.174 Mean :1.05 Mean :0.6704 Mean :4.696
## 3rd Qu.:4.000 3rd Qu.:1.60 3rd Qu.:1.0000 3rd Qu.:7.000
## Max. :4.000 Max. :6.20 Max. :3.0000 Max. :7.000
## Sex Max.HR
## Min. :0.0000 Min. : 71.0
## 1st Qu.:0.0000 1st Qu.:133.0
## Median :1.0000 Median :153.5
## Mean :0.6778 Mean :149.7
## 3rd Qu.:1.0000 3rd Qu.:166.0
## Max. :1.0000 Max. :202.0The na.omit function is used to remove the the rows that are containing with N/A values which helps in more accurate prediction.
Setting up the model
As The outcome is binary i.e the disease -Present in the body (or) absent in the body .I would be using the logistic regression approach .Since it gives the probability of the occurrence of the event based on the independent variables .The most of the variables i filtered are independent variables.
The data model i would be going to examine is
Building models
I am going to build four different models.
The first model contains all the parameters present in the table.
The second model contains the elements that I am interested in examining with the heart rate.
The third model is the filtered variables of the second model.
The fourth model contains the most accurate parameters by examining the above three models.
The prior intercept That I am considering is the range of the heartbeat. The data on the range of the heartbeat is gathered by analyzing the summary function. Hence the prior intercept I selected is (71,65.5) because the heartbeat is ranging from 71 to 202.
ggpairs(dataset1)
plot_normal(mean = 71,sd=65.5)
The explanatory analysis is performed by using the ggpairs function ,where in which the correlation o fthe heart beat is negative with most of the parameters I considered.
Implementing the model
Model-1(Full model)
Implement a full model, including all the relevant variables as predictors. If you have chosen to exclude any variables, explain why.
I am assuming that(Age+BP+FBS.over.120+Cholesterol+Chest.pain.type+ST.depression+Number.of.vessels.fluro+Thallium+Sex+Max.HR) as predictors and all the predictor variables are not statistically significant with each other .I am going to set up the heart beat range as the prior_intercept while the prior normal as the week prior normal .Hence it is set as auto scale.
model_1 <- stan_glm(
Max.HR~.,
data = dataset1, family = gaussian,
prior_intercept = normal(71,65.5),
prior = normal(0, 1, autoscale = TRUE),
prior_aux = exponential(1, autoscale = TRUE),
chains = 4, iter = 5000*2, seed = 84735,prior_PD = TRUE)##
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## Chain 4:summary(model_1)##
## Model Info:
## function: stan_glm
## family: gaussian [identity]
## formula: Max.HR ~ .
## algorithm: sampling
## sample: 20000 (posterior sample size)
## priors: see help('prior_summary')
## observations: 270
## predictors: 10
##
## Estimates:
## mean sd 10% 50% 90%
## (Intercept) 68.4 277.4 -285.4 67.2 422.6
## Age 0.0 2.6 -3.3 0.0 3.3
## BP 0.0 1.3 -1.6 0.0 1.7
## FBS.over.120 -0.8 65.2 -84.3 -0.4 83.0
## Cholesterol 0.0 0.4 -0.6 0.0 0.6
## Chest.pain.type -0.1 24.2 -31.3 0.1 31.0
## ST.depression 0.0 20.4 -26.1 -0.1 26.1
## Number.of.vessels.fluro 0.0 24.4 -31.7 0.0 31.7
## Thallium 0.0 12.0 -15.4 -0.1 15.4
## Sex -0.2 49.6 -63.6 0.0 62.9
## sigma 23.2 22.9 2.4 16.2 53.2
##
## MCMC diagnostics
## mcse Rhat n_eff
## (Intercept) 1.8 1.0 24293
## Age 0.0 1.0 24868
## BP 0.0 1.0 23988
## FBS.over.120 0.4 1.0 23997
## Cholesterol 0.0 1.0 25961
## Chest.pain.type 0.2 1.0 23869
## ST.depression 0.1 1.0 22375
## Number.of.vessels.fluro 0.2 1.0 24384
## Thallium 0.1 1.0 24270
## Sex 0.3 1.0 23243
## sigma 0.1 1.0 27181
## log-posterior 0.0 1.0 10360
##
## For each parameter, mcse is Monte Carlo standard error, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence Rhat=1).The summary of the function determines that the prior intercept is with the mean of 68.4 and the standard deviation 277.4, with the value of 422.6 in 90% confidence interval.The rhat value is 1 for all the predictor variables.
Check the reliability of the MCMC simulations by reporting appropriate graphical and numerical measures and interpreting them.
mcmc_trace(model_1, size = 0.1)
mcmc_dens_overlay(model_1)
mcmc_acf(model_1)
neff_ratio(model_1)## (Intercept) Age BP
## 1.21465 1.24340 1.19940
## FBS.over.120 Cholesterol Chest.pain.type
## 1.19985 1.29805 1.19345
## ST.depression Number.of.vessels.fluro Thallium
## 1.11875 1.21920 1.21350
## Sex sigma
## 1.16215 1.35905rhat(model_1)## (Intercept) Age BP
## 1.0000148 0.9998946 0.9998509
## FBS.over.120 Cholesterol Chest.pain.type
## 1.0000517 1.0001498 1.0002919
## ST.depression Number.of.vessels.fluro Thallium
## 0.9999149 1.0001425 0.9999348
## Sex sigma
## 0.9998162 0.9999887The mcmc_trance function is used to denote the chains .The chains produced for the above model appears to be mixed well together .Hence they are considered as the stable.
The mcmc overlay function represents the probability density function of each parameter overlap with each other well .In the above graph papers that like sufficiently overlapped with each other.
By diagnosing the Rhat values it clearly appears that Rhat values are equal to 1 in all the cases above indicating that there are no issues with autocorrelation .
prior_summary(model_1)## Priors for model 'model_1'
## ------
## Intercept (after predictors centered)
## ~ normal(location = 71, scale = 66)
##
## Coefficients
## Specified prior:
## ~ normal(location = [0,0,0,...], scale = [1,1,1,...])
## Adjusted prior:
## ~ normal(location = [0,0,0,...], scale = [ 2.54, 1.30,65.09,...])
##
## Auxiliary (sigma)
## Specified prior:
## ~ exponential(rate = 1)
## Adjusted prior:
## ~ exponential(rate = 0.043)
## ------
## See help('prior_summary.stanreg') for more detailsInterpret the results of the model’s parameters’ PDFs - explain whether is support for the hypotheses you posited earlier, one hypothesis at a time, by connecting the results to the hypothesis’ specification.
The prior_summary function gives the summary of the prior of the specified model .As the summary function suggests that most of the functions I considered is mostly true.The hypothesis I considered for the prior summary is mostly turned out to be true since the values of the prior normal did not vary drastically.
set.seed(84735)
posteriorpredict=posterior_predict(model_1)posterior_interval(posteriorpredict,prob=0.95)## 2.5% 97.5%
## 1 -158.00778 295.6741
## 2 -262.96589 407.3006
## 3 -102.61325 243.8741
## 4 -105.34161 245.0872
## 5 -135.33827 274.6445
## 6 -119.29196 258.8661
## 7 -114.19625 256.1693
## 8 -103.38281 244.5444
## 9 -111.69581 255.4053
## 10 -213.45535 357.4354
## 11 -97.40839 238.3196
## 12 -101.44728 247.5567
## 13 -96.20570 241.2249
## 14 -136.92320 278.8759
## 15 -107.35876 250.8213
## 16 -144.09746 283.7585
## 17 -120.83819 262.3766
## 18 -149.35894 287.0005
## 19 -134.71715 275.6778
## 20 -139.50415 285.8933
## 21 -128.18235 270.5599
## 22 -98.25602 241.0189
## 23 -111.33449 250.0054
## 24 -109.85958 251.0453
## 25 -139.89956 283.2596
## 26 -101.18156 244.1473
## 27 -109.19281 251.4002
## 28 -104.18048 242.6526
## 29 -108.69380 249.6187
## 30 -150.32866 292.2687
## 31 -90.06356 230.6737
## 32 -120.44997 261.6519
## 33 -124.76692 267.5292
## 34 -157.50361 297.1136
## 35 -101.40018 245.9790
## 36 -141.59009 281.3582
## 37 -103.40836 246.8978
## 38 -133.34895 277.3536
## 39 -110.05277 252.3191
## 40 -99.42348 241.1643
## 41 -122.21979 263.8389
## 42 -111.56784 255.8134
## 43 -92.72775 236.6822
## 44 -155.65853 300.5842
## 45 -125.08360 265.6709
## 46 -126.85289 268.9268
## 47 -140.55250 285.2587
## 48 -113.04301 253.7173
## 49 -146.75545 290.2476
## 50 -148.34737 292.1959
## 51 -113.89767 252.8348
## 52 -136.64040 276.7356
## 53 -183.21926 325.4019
## 54 -128.40098 270.1167
## 55 -111.10027 252.2010
## 56 -133.81035 277.8151
## 57 -137.07644 276.9251
## 58 -143.08642 285.8389
## 59 -135.15238 279.4056
## 60 -106.79296 247.7315
## 61 -161.04510 297.2959
## 62 -111.17450 251.9221
## 63 -99.91699 241.4805
## 64 -133.86536 276.2721
## 65 -157.06393 298.2776
## 66 -95.46118 238.7804
## 67 -97.01244 238.4794
## 68 -152.63902 298.8871
## 69 -108.14546 246.7350
## 70 -106.23359 243.3181
## 71 -128.48127 271.9234
## 72 -126.08874 267.2351
## 73 -126.12001 270.3636
## 74 -132.31979 277.0207
## 75 -133.47540 272.4917
## 76 -142.06893 283.6357
## 77 -130.35565 271.2062
## 78 -140.07832 283.6633
## 79 -115.88258 257.3948
## 80 -96.94956 238.0086
## 81 -104.36789 245.7848
## 82 -125.72276 268.9277
## 83 -101.22634 244.3718
## 84 -106.02672 246.9353
## 85 -102.98021 244.7546
## 86 -134.77254 277.9129
## 87 -137.99796 276.6721
## 88 -189.73272 334.8687
## 89 -126.19559 265.9252
## 90 -110.23564 253.8595
## 91 -101.93326 244.8761
## 92 -120.97490 261.2569
## 93 -102.39254 242.6494
## 94 -106.29853 245.1135
## 95 -111.33541 250.6213
## 96 -100.75621 238.7396
## 97 -110.13505 252.4161
## 98 -114.33470 255.5258
## 99 -118.12936 265.0873
## 100 -107.07551 247.9782
## 101 -133.94802 281.0749
## 102 -105.49460 247.6201
## 103 -106.22783 248.2406
## 104 -175.13568 315.7617
## 105 -98.43142 238.5458
## 106 -99.13581 240.7373
## 107 -109.61091 247.8879
## 108 -131.33399 273.7565
## 109 -112.83212 252.9391
## 110 -137.65195 280.6980
## 111 -169.89244 314.5338
## 112 -118.30835 260.8246
## 113 -124.24891 269.0011
## 114 -136.92295 279.9930
## 115 -115.31642 256.4062
## 116 -111.64506 252.8278
## 117 -99.40387 241.9286
## 118 -231.11345 373.4977
## 119 -149.07308 293.1604
## 120 -131.54012 273.0641
## 121 -140.09030 276.2120
## 122 -126.61408 267.7680
## 123 -99.73624 243.4220
## 124 -142.22985 283.8270
## 125 -83.63741 224.3571
## 126 -117.99212 259.9401
## 127 -114.75119 257.0612
## 128 -104.96657 247.9505
## 129 -96.79779 236.3482
## 130 -138.40548 277.8545
## 131 -126.66212 268.5875
## 132 -113.33220 253.9155
## 133 -106.95384 249.6328
## 134 -107.64303 246.5552
## 135 -96.58178 240.0884
## 136 -113.21846 255.6189
## 137 -121.05203 261.3707
## 138 -118.62741 262.7470
## 139 -135.35587 279.0530
## 140 -103.36632 244.1865
## 141 -117.25565 257.4153
## 142 -97.33564 239.5435
## 143 -90.17701 231.0615
## 144 -118.12133 260.9010
## 145 -157.79674 306.1373
## 146 -112.05480 253.6583
## 147 -100.19824 242.1300
## 148 -132.30123 274.2670
## 149 -124.07981 265.7279
## 150 -118.65862 259.6339
## 151 -108.12064 250.6074
## 152 -100.71770 244.4160
## 153 -108.99249 249.5901
## 154 -126.91450 266.9694
## 155 -94.60612 236.9024
## 156 -129.59110 267.7707
## 157 -178.75146 315.1195
## 158 -112.51282 252.1901
## 159 -131.88890 273.6052
## 160 -184.20472 326.9156
## 161 -169.09611 306.3836
## 162 -110.82357 254.4854
## 163 -97.83054 239.0752
## 164 -121.34893 264.0650
## 165 -116.50927 260.9838
## 166 -158.95517 304.0637
## 167 -113.95057 254.3329
## 168 -124.20801 265.3512
## 169 -108.94255 250.2153
## 170 -150.63749 295.2180
## 171 -167.41814 311.6040
## 172 -144.01807 284.3545
## 173 -135.68069 274.5731
## 174 -113.47055 252.7420
## 175 -155.05892 297.8833
## 176 -168.85069 306.7464
## 177 -173.39853 309.9416
## 178 -127.43010 267.7891
## 179 -149.08537 286.8923
## 180 -95.79632 241.9892
## 181 -132.20448 273.9267
## 182 -163.86524 307.6870
## 183 -131.16008 272.1930
## 184 -125.49248 266.5666
## 185 -149.46268 292.4159
## 186 -102.29407 245.7991
## 187 -101.65422 242.3243
## 188 -168.38237 305.5580
## 189 -146.30324 290.4313
## 190 -136.31441 275.3164
## 191 -89.78971 232.2468
## 192 -133.33675 277.0037
## 193 -120.96203 261.5370
## 194 -125.17336 268.3648
## 195 -133.62069 274.0111
## 196 -103.30542 247.4296
## 197 -121.17622 258.1054
## 198 -109.57962 247.8009
## 199 -147.50503 289.4001
## 200 -162.51026 302.8358
## 201 -108.03887 249.8694
## 202 -117.57413 255.7281
## 203 -110.95340 251.7274
## 204 -105.22947 242.9276
## 205 -110.42003 253.3404
## 206 -157.16114 305.0424
## 207 -124.65226 265.9055
## 208 -102.04007 245.4607
## 209 -110.49630 251.9335
## 210 -133.02795 275.9990
## 211 -147.68283 291.9713
## 212 -127.31355 268.5910
## 213 -110.66077 253.4143
## 214 -134.40061 274.3521
## 215 -141.90970 287.3261
## 216 -118.80678 263.6353
## 217 -105.29829 246.5081
## 218 -118.04179 259.3919
## 219 -94.50680 234.6299
## 220 -109.12112 251.9488
## 221 -115.88336 255.1391
## 222 -116.79995 257.7387
## 223 -107.43944 251.3894
## 224 -172.83308 313.0805
## 225 -132.69839 278.7157
## 226 -110.53615 256.9472
## 227 -101.94209 244.5311
## 228 -169.02406 310.3664
## 229 -159.56561 302.8405
## 230 -126.92524 272.5028
## 231 -118.25432 259.3952
## 232 -115.22892 256.7401
## 233 -105.23800 245.8092
## 234 -98.11144 238.5409
## 235 -130.80913 269.7785
## 236 -242.08613 378.9950
## 237 -104.68283 249.1427
## 238 -125.22108 267.2686
## 239 -92.05316 236.2209
## 240 -113.21391 252.3610
## 241 -170.82036 315.8545
## 242 -116.07081 259.4851
## 243 -99.23439 246.9751
## 244 -141.58340 285.0544
## 245 -102.55494 244.6811
## 246 -95.59278 237.6724
## 247 -114.80435 257.2240
## 248 -113.39979 256.4642
## 249 -115.70479 259.3808
## 250 -125.81888 268.0905
## 251 -101.73158 247.3796
## 252 -107.41434 249.2960
## 253 -109.76311 251.7733
## 254 -108.21695 249.2987
## 255 -131.97374 271.5275
## 256 -150.20478 292.1436
## 257 -127.97777 267.9701
## 258 -119.36366 256.7650
## 259 -114.37555 256.5720
## 260 -102.99655 244.6851
## 261 -119.48619 257.3255
## 262 -117.08806 258.9993
## 263 -103.21746 244.2117
## 264 -96.25070 237.8444
## 265 -110.59478 251.5750
## 266 -153.71657 295.7314
## 267 -113.64440 255.5304
## 268 -108.59156 252.4842
## 269 -97.88429 240.1845
## 270 -149.45669 290.4658summary(posteriorpredict)## 1 2 3 4
## Min. :-411.197 Min. :-617.63 Min. :-333.36 Min. :-320.94
## 1st Qu.: -7.231 1st Qu.: -44.27 1st Qu.: 12.80 1st Qu.: 11.93
## Median : 69.922 Median : 72.25 Median : 71.31 Median : 71.57
## Mean : 70.429 Mean : 71.59 Mean : 71.20 Mean : 71.12
## 3rd Qu.: 148.322 3rd Qu.: 189.47 3rd Qu.: 129.93 3rd Qu.: 129.68
## Max. : 590.500 Max. : 765.19 Max. : 463.30 Max. : 491.31
## 5 6 7 8
## Min. :-348.573 Min. :-540.983 Min. :-449.383 Min. :-388.63
## 1st Qu.: 1.511 1st Qu.: 7.226 1st Qu.: 7.229 1st Qu.: 12.31
## Median : 72.068 Median : 69.436 Median : 70.543 Median : 70.89
## Mean : 71.283 Mean : 70.491 Mean : 69.962 Mean : 70.74
## 3rd Qu.: 140.406 3rd Qu.: 133.819 3rd Qu.: 132.388 3rd Qu.: 128.83
## Max. : 627.325 Max. : 561.612 Max. : 726.729 Max. : 495.75
## 9 10 11 12
## Min. :-409.860 Min. :-531.63 Min. :-403.06 Min. :-347.04
## 1st Qu.: 8.981 1st Qu.: -27.35 1st Qu.: 13.80 1st Qu.: 12.07
## Median : 71.477 Median : 72.98 Median : 70.63 Median : 70.67
## Mean : 71.144 Mean : 71.45 Mean : 70.72 Mean : 71.28
## 3rd Qu.: 133.273 3rd Qu.: 169.91 3rd Qu.: 128.04 3rd Qu.: 130.00
## Max. : 588.653 Max. : 643.42 Max. : 460.00 Max. : 478.19
## 13 14 15 16
## Min. :-447.62 Min. :-331.6175 Min. :-409.88 Min. :-336.007
## 1st Qu.: 12.88 1st Qu.: 0.6916 1st Qu.: 11.41 1st Qu.: -2.246
## Median : 71.20 Median : 72.3307 Median : 71.38 Median : 70.701
## Mean : 71.29 Mean : 71.7353 Mean : 71.53 Mean : 71.034
## 3rd Qu.: 129.07 3rd Qu.: 141.8790 3rd Qu.: 132.46 3rd Qu.: 145.117
## Max. : 435.63 Max. : 493.2132 Max. : 467.06 Max. : 492.574
## 17 18 19 20
## Min. :-488.453 Min. :-422.517 Min. :-490.368 Min. :-403.8621
## 1st Qu.: 6.586 1st Qu.: -4.293 1st Qu.: 1.472 1st Qu.: -0.5977
## Median : 71.238 Median : 69.876 Median : 71.539 Median : 71.2935
## Mean : 71.255 Mean : 70.111 Mean : 70.790 Mean : 71.3837
## 3rd Qu.: 136.584 3rd Qu.: 143.934 3rd Qu.: 141.029 3rd Qu.: 142.6228
## Max. : 501.219 Max. : 594.672 Max. : 489.602 Max. : 589.1934
## 21 22 23 24
## Min. :-520.463 Min. :-492.67 Min. :-355.060 Min. :-384.286
## 1st Qu.: 2.753 1st Qu.: 13.02 1st Qu.: 9.186 1st Qu.: 9.427
## Median : 71.443 Median : 71.44 Median : 70.320 Median : 70.164
## Mean : 70.355 Mean : 71.44 Mean : 70.646 Mean : 70.669
## 3rd Qu.: 137.795 3rd Qu.: 129.21 3rd Qu.: 133.497 3rd Qu.: 131.458
## Max. : 474.692 Max. : 483.83 Max. : 545.250 Max. : 522.441
## 25 26 27 28
## Min. :-402.1486 Min. :-354.64 Min. :-409.57 Min. :-402.01
## 1st Qu.: 0.3878 1st Qu.: 13.25 1st Qu.: 10.14 1st Qu.: 12.20
## Median : 70.2167 Median : 71.55 Median : 71.72 Median : 70.97
## Mean : 70.9480 Mean : 71.36 Mean : 71.28 Mean : 71.25
## 3rd Qu.: 141.2706 3rd Qu.: 130.24 3rd Qu.: 132.83 3rd Qu.: 130.56
## Max. : 553.2390 Max. : 632.23 Max. : 567.23 Max. : 513.11
## 29 30 31 32
## Min. :-405.64 Min. :-513.713 Min. :-361.43 Min. :-453.288
## 1st Qu.: 10.56 1st Qu.: -6.742 1st Qu.: 16.14 1st Qu.: 5.037
## Median : 71.72 Median : 69.892 Median : 70.69 Median : 71.316
## Mean : 70.96 Mean : 69.968 Mean : 70.76 Mean : 71.088
## 3rd Qu.: 130.56 3rd Qu.: 146.280 3rd Qu.: 125.86 3rd Qu.: 135.929
## Max. : 470.66 Max. : 617.303 Max. : 537.98 Max. : 578.401
## 33 34 35 36
## Min. :-355.563 Min. :-438.287 Min. :-359.33 Min. :-403.3822
## 1st Qu.: 3.806 1st Qu.: -5.749 1st Qu.: 11.83 1st Qu.: -0.4558
## Median : 71.320 Median : 72.069 Median : 69.98 Median : 71.2452
## Mean : 70.898 Mean : 71.362 Mean : 70.95 Mean : 70.3392
## 3rd Qu.: 137.476 3rd Qu.: 149.108 3rd Qu.: 129.61 3rd Qu.: 140.5517
## Max. : 585.841 Max. : 730.120 Max. : 485.59 Max. : 483.6220
## 37 38 39 40
## Min. :-366.78 Min. :-523.538 Min. :-424.419 Min. :-334.73
## 1st Qu.: 11.93 1st Qu.: 1.813 1st Qu.: 8.524 1st Qu.: 12.44
## Median : 70.84 Median : 71.481 Median : 70.349 Median : 70.67
## Mean : 70.95 Mean : 71.668 Mean : 70.511 Mean : 70.85
## 3rd Qu.: 130.58 3rd Qu.: 141.511 3rd Qu.: 132.526 3rd Qu.: 129.05
## Max. : 490.93 Max. : 489.918 Max. : 468.996 Max. : 489.35
## 41 42 43 44
## Min. :-485.158 Min. :-316.435 Min. :-283.92 Min. :-482.733
## 1st Qu.: 5.504 1st Qu.: 8.755 1st Qu.: 14.75 1st Qu.: -8.157
## Median : 70.975 Median : 71.852 Median : 71.33 Median : 70.134
## Mean : 71.247 Mean : 71.255 Mean : 71.10 Mean : 69.618
## 3rd Qu.: 137.350 3rd Qu.: 134.427 3rd Qu.: 127.16 3rd Qu.: 146.611
## Max. : 474.642 Max. : 459.779 Max. : 466.51 Max. : 572.136
## 45 46 47 48
## Min. :-341.544 Min. :-350.30 Min. :-367.982 Min. :-347.831
## 1st Qu.: 3.386 1st Qu.: 3.03 1st Qu.: -1.471 1st Qu.: 7.525
## Median : 69.561 Median : 69.68 Median : 71.112 Median : 70.102
## Mean : 70.034 Mean : 70.59 Mean : 70.506 Mean : 70.325
## 3rd Qu.: 135.720 3rd Qu.: 137.51 3rd Qu.: 142.549 3rd Qu.: 132.646
## Max. : 494.134 Max. : 470.36 Max. : 523.830 Max. : 458.801
## 49 50 51 52
## Min. :-383.909 Min. :-501.219 Min. :-384.905 Min. :-388.4070
## 1st Qu.: -4.993 1st Qu.: -3.877 1st Qu.: 8.454 1st Qu.: 0.3531
## Median : 72.328 Median : 71.098 Median : 71.046 Median : 69.7409
## Mean : 71.656 Mean : 71.032 Mean : 71.157 Mean : 70.1362
## 3rd Qu.: 148.482 3rd Qu.: 146.439 3rd Qu.: 134.050 3rd Qu.: 139.8470
## Max. : 571.978 Max. : 480.598 Max. : 450.069 Max. : 566.7491
## 53 54 55 56
## Min. :-525.93 Min. :-411.341 Min. :-366.286 Min. :-356.5295
## 1st Qu.: -15.80 1st Qu.: 3.648 1st Qu.: 9.606 1st Qu.: 0.5407
## Median : 71.27 Median : 71.136 Median : 72.086 Median : 70.2662
## Mean : 70.95 Mean : 71.301 Mean : 71.646 Mean : 70.6473
## 3rd Qu.: 158.20 3rd Qu.: 139.025 3rd Qu.: 133.571 3rd Qu.: 140.4157
## Max. : 581.04 Max. : 535.220 Max. : 578.476 Max. : 559.6981
## 57 58 59 60
## Min. :-534.396 Min. :-488.533 Min. :-338.5381 Min. :-528.27
## 1st Qu.: 1.877 1st Qu.: -4.279 1st Qu.: 0.5888 1st Qu.: 11.25
## Median : 69.919 Median : 69.912 Median : 72.4543 Median : 71.01
## Mean : 70.447 Mean : 70.035 Mean : 71.8368 Mean : 71.05
## 3rd Qu.: 140.047 3rd Qu.: 144.439 3rd Qu.: 142.5730 3rd Qu.: 130.74
## Max. : 584.697 Max. : 558.006 Max. : 509.7292 Max. : 636.13
## 61 62 63 64
## Min. :-489.548 Min. :-354.95 Min. :-333.15 Min. :-364.984
## 1st Qu.: -6.441 1st Qu.: 9.45 1st Qu.: 11.29 1st Qu.: 2.063
## Median : 69.725 Median : 70.99 Median : 71.49 Median : 71.662
## Mean : 70.093 Mean : 70.97 Mean : 70.61 Mean : 71.886
## 3rd Qu.: 147.396 3rd Qu.: 131.98 3rd Qu.: 128.92 3rd Qu.: 141.093
## Max. : 553.789 Max. : 575.54 Max. : 530.82 Max. : 685.179
## 65 66 67 68
## Min. :-366.722 Min. :-382.53 Min. :-397.32 Min. :-482.641
## 1st Qu.: -7.481 1st Qu.: 14.30 1st Qu.: 14.68 1st Qu.: -5.335
## Median : 69.598 Median : 71.25 Median : 71.37 Median : 70.498
## Mean : 70.668 Mean : 71.25 Mean : 71.18 Mean : 71.291
## 3rd Qu.: 149.685 3rd Qu.: 127.86 3rd Qu.: 128.33 3rd Qu.: 147.982
## Max. : 598.621 Max. : 496.84 Max. : 536.49 Max. : 623.833
## 69 70 71 72
## Min. :-396.71 Min. :-353.52 Min. :-429.923 Min. :-346.682
## 1st Qu.: 10.44 1st Qu.: 11.45 1st Qu.: 2.138 1st Qu.: 3.968
## Median : 71.48 Median : 70.84 Median : 70.862 Median : 72.031
## Mean : 70.99 Mean : 70.43 Mean : 70.647 Mean : 71.186
## 3rd Qu.: 131.78 3rd Qu.: 129.62 3rd Qu.: 137.913 3rd Qu.: 138.327
## Max. : 605.65 Max. : 492.01 Max. : 540.051 Max. : 474.376
## 73 74 75 76
## Min. :-311.800 Min. :-454.863 Min. :-356.206 Min. :-380.907
## 1st Qu.: 4.835 1st Qu.: 1.331 1st Qu.: 1.495 1st Qu.: -3.752
## Median : 70.893 Median : 70.797 Median : 70.669 Median : 71.724
## Mean : 71.862 Mean : 71.439 Mean : 70.614 Mean : 71.208
## 3rd Qu.: 139.147 3rd Qu.: 140.970 3rd Qu.: 139.871 3rd Qu.: 145.860
## Max. : 547.002 Max. : 553.516 Max. : 537.455 Max. : 552.658
## 77 78 79 80
## Min. :-372.61 Min. :-503.009 Min. :-316.346 Min. :-334.10
## 1st Qu.: 1.77 1st Qu.: -1.817 1st Qu.: 7.193 1st Qu.: 14.02
## Median : 70.40 Median : 70.623 Median : 71.730 Median : 70.20
## Mean : 70.23 Mean : 70.940 Mean : 71.029 Mean : 70.60
## 3rd Qu.: 139.26 3rd Qu.: 143.046 3rd Qu.: 134.767 3rd Qu.: 127.34
## Max. : 466.37 Max. : 502.400 Max. : 594.330 Max. : 427.01
## 81 82 83 84
## Min. :-512.97 Min. :-377.256 Min. :-306.73 Min. :-408.221
## 1st Qu.: 12.11 1st Qu.: 2.996 1st Qu.: 12.50 1st Qu.: 9.966
## Median : 70.59 Median : 69.642 Median : 71.57 Median : 71.185
## Mean : 70.99 Mean : 70.629 Mean : 71.48 Mean : 70.874
## 3rd Qu.: 129.68 3rd Qu.: 137.764 3rd Qu.: 130.08 3rd Qu.: 131.980
## Max. : 488.05 Max. : 548.832 Max. : 619.13 Max. : 555.700
## 85 86 87 88
## Min. :-324.38 Min. :-362.3839 Min. :-421.502 Min. :-407.13
## 1st Qu.: 11.53 1st Qu.: 0.2786 1st Qu.: -1.011 1st Qu.: -17.87
## Median : 70.06 Median : 70.4060 Median : 70.615 Median : 72.31
## Mean : 70.19 Mean : 71.3721 Mean : 70.303 Mean : 71.74
## 3rd Qu.: 128.54 3rd Qu.: 142.5845 3rd Qu.: 141.637 3rd Qu.: 160.84
## Max. : 561.79 Max. : 550.9005 Max. : 502.822 Max. : 612.57
## 89 90 91 92
## Min. :-391.733 Min. :-387.84 Min. :-412.66 Min. :-443.856
## 1st Qu.: 4.008 1st Qu.: 8.08 1st Qu.: 13.03 1st Qu.: 6.534
## Median : 71.379 Median : 71.13 Median : 70.98 Median : 72.065
## Mean : 71.212 Mean : 71.38 Mean : 71.50 Mean : 71.199
## 3rd Qu.: 139.364 3rd Qu.: 134.61 3rd Qu.: 130.17 3rd Qu.: 135.859
## Max. : 502.996 Max. : 506.07 Max. : 445.96 Max. : 579.682
## 93 94 95 96
## Min. :-306.31 Min. :-333.15 Min. :-504.44 Min. :-339.97
## 1st Qu.: 12.84 1st Qu.: 10.87 1st Qu.: 10.38 1st Qu.: 11.55
## Median : 70.60 Median : 70.55 Median : 71.49 Median : 70.33
## Mean : 70.57 Mean : 70.07 Mean : 70.56 Mean : 70.19
## 3rd Qu.: 128.17 3rd Qu.: 129.90 3rd Qu.: 131.14 3rd Qu.: 129.02
## Max. : 460.96 Max. : 517.34 Max. : 537.53 Max. : 493.69
## 97 98 99 100
## Min. :-380.50 Min. :-459.756 Min. :-385.889 Min. :-362.05
## 1st Qu.: 10.89 1st Qu.: 8.296 1st Qu.: 7.675 1st Qu.: 10.54
## Median : 71.30 Median : 70.806 Median : 72.657 Median : 71.56
## Mean : 71.48 Mean : 70.423 Mean : 72.213 Mean : 71.16
## 3rd Qu.: 132.44 3rd Qu.: 132.775 3rd Qu.: 136.832 3rd Qu.: 131.60
## Max. : 734.19 Max. : 596.176 Max. : 514.875 Max. : 480.36
## 101 102 103 104
## Min. :-398.3221 Min. :-419.59 Min. :-314.50 Min. :-490.85
## 1st Qu.: -0.5229 1st Qu.: 11.49 1st Qu.: 10.94 1st Qu.: -12.95
## Median : 70.8834 Median : 69.72 Median : 70.58 Median : 71.28
## Mean : 70.6739 Mean : 70.66 Mean : 70.97 Mean : 70.59
## 3rd Qu.: 140.2295 3rd Qu.: 129.61 3rd Qu.: 131.44 3rd Qu.: 153.43
## Max. : 621.4519 Max. : 469.71 Max. : 652.23 Max. : 545.65
## 105 106 107 108
## Min. :-344.75 Min. :-309.80 Min. :-504.10 Min. :-404.714
## 1st Qu.: 13.59 1st Qu.: 13.19 1st Qu.: 10.03 1st Qu.: 2.237
## Median : 71.38 Median : 70.41 Median : 70.43 Median : 72.226
## Mean : 71.12 Mean : 70.89 Mean : 70.21 Mean : 71.452
## 3rd Qu.: 128.98 3rd Qu.: 129.21 3rd Qu.: 130.32 3rd Qu.: 140.289
## Max. : 653.50 Max. : 665.86 Max. : 442.24 Max. : 487.390
## 109 110 111 112
## Min. :-348.937 Min. :-375.9020 Min. :-439.46 Min. :-370.93
## 1st Qu.: 7.994 1st Qu.: 0.1404 1st Qu.: -11.59 1st Qu.: 4.49
## Median : 69.874 Median : 70.7958 Median : 72.25 Median : 70.83
## Mean : 70.468 Mean : 70.5872 Mean : 71.72 Mean : 70.43
## 3rd Qu.: 132.403 3rd Qu.: 140.3446 3rd Qu.: 154.36 3rd Qu.: 135.74
## Max. : 550.462 Max. : 510.4342 Max. : 504.65 Max. : 477.66
## 113 114 115 116
## Min. :-444.74 Min. :-327.4352 Min. :-306.692 Min. :-354.461
## 1st Qu.: 6.22 1st Qu.: 0.0492 1st Qu.: 6.247 1st Qu.: 9.384
## Median : 70.80 Median : 70.5386 Median : 70.347 Median : 71.914
## Mean : 71.97 Mean : 70.8711 Mean : 70.839 Mean : 71.736
## 3rd Qu.: 137.84 3rd Qu.: 141.2629 3rd Qu.: 135.293 3rd Qu.: 133.621
## Max. : 471.19 Max. : 546.2123 Max. : 607.728 Max. : 446.065
## 117 118 119 120
## Min. :-336.72 Min. :-595.29 Min. :-749.534 Min. :-348.44
## 1st Qu.: 12.48 1st Qu.: -30.37 1st Qu.: -4.022 1st Qu.: 1.84
## Median : 70.04 Median : 72.05 Median : 71.810 Median : 69.75
## Mean : 70.50 Mean : 71.66 Mean : 71.912 Mean : 69.86
## 3rd Qu.: 128.20 3rd Qu.: 174.05 3rd Qu.: 147.504 3rd Qu.: 138.24
## Max. : 503.74 Max. : 715.62 Max. : 516.583 Max. : 520.82
## 121 122 123 124
## Min. :-356.3366 Min. :-443.421 Min. :-339.27 Min. :-539.9084
## 1st Qu.: -0.1928 1st Qu.: 3.447 1st Qu.: 12.54 1st Qu.: -0.2374
## Median : 71.5712 Median : 70.875 Median : 71.37 Median : 71.8033
## Mean : 70.5604 Mean : 70.590 Mean : 71.57 Mean : 72.2391
## 3rd Qu.: 141.6608 3rd Qu.: 137.633 3rd Qu.: 130.76 3rd Qu.: 145.6020
## Max. : 513.1220 Max. : 456.201 Max. : 546.11 Max. : 643.7576
## 125 126 127 128
## Min. :-377.87 Min. :-306.472 Min. :-366.364 Min. :-426.00
## 1st Qu.: 18.14 1st Qu.: 6.678 1st Qu.: 7.813 1st Qu.: 10.93
## Median : 71.53 Median : 72.207 Median : 71.631 Median : 71.47
## Mean : 70.98 Mean : 71.828 Mean : 70.867 Mean : 71.30
## 3rd Qu.: 124.34 3rd Qu.: 135.634 3rd Qu.: 133.269 3rd Qu.: 131.86
## Max. : 439.19 Max. : 483.493 Max. : 773.975 Max. : 472.70
## 129 130 131 132
## Min. :-405.28 Min. :-410.809 Min. :-447.548 Min. :-525.053
## 1st Qu.: 13.90 1st Qu.: -2.151 1st Qu.: 4.487 1st Qu.: 8.763
## Median : 71.06 Median : 70.377 Median : 70.917 Median : 71.412
## Mean : 71.01 Mean : 69.869 Mean : 70.924 Mean : 70.406
## 3rd Qu.: 128.10 3rd Qu.: 141.523 3rd Qu.: 137.719 3rd Qu.: 132.169
## Max. : 416.26 Max. : 652.394 Max. : 496.210 Max. : 887.240
## 133 134 135 136
## Min. :-356.216 Min. :-337.16 Min. :-269.03 Min. :-528.112
## 1st Qu.: 9.713 1st Qu.: 12.45 1st Qu.: 13.30 1st Qu.: 7.375
## Median : 71.034 Median : 70.62 Median : 70.74 Median : 71.764
## Mean : 71.163 Mean : 70.59 Mean : 71.13 Mean : 71.257
## 3rd Qu.: 132.347 3rd Qu.: 129.70 3rd Qu.: 129.23 3rd Qu.: 135.735
## Max. : 434.524 Max. : 515.72 Max. : 483.80 Max. : 434.346
## 137 138 139 140
## Min. :-374.843 Min. :-381.624 Min. :-407.5262 Min. :-372.51
## 1st Qu.: 8.007 1st Qu.: 5.852 1st Qu.: 0.1344 1st Qu.: 11.74
## Median : 71.863 Median : 69.377 Median : 71.2656 Median : 70.44
## Mean : 71.876 Mean : 69.926 Mean : 71.0722 Mean : 70.58
## 3rd Qu.: 136.482 3rd Qu.: 135.434 3rd Qu.: 142.4424 3rd Qu.: 129.09
## Max. : 555.726 Max. : 487.293 Max. : 730.2026 Max. : 613.72
## 141 142 143 144
## Min. :-440.513 Min. :-340.62 Min. :-499.63 Min. :-317.425
## 1st Qu.: 8.214 1st Qu.: 14.34 1st Qu.: 16.99 1st Qu.: 7.079
## Median : 71.770 Median : 71.14 Median : 70.81 Median : 70.854
## Mean : 71.002 Mean : 71.23 Mean : 71.03 Mean : 70.983
## 3rd Qu.: 133.896 3rd Qu.: 127.61 3rd Qu.: 125.83 3rd Qu.: 135.405
## Max. : 595.338 Max. : 422.61 Max. : 428.62 Max. : 530.456
## 145 146 147 148
## Min. :-490.759 Min. :-368.962 Min. :-345.52 Min. :-353.207
## 1st Qu.: -9.257 1st Qu.: 8.781 1st Qu.: 12.77 1st Qu.: 0.845
## Median : 72.037 Median : 70.875 Median : 70.75 Median : 69.230
## Mean : 72.178 Mean : 70.364 Mean : 70.82 Mean : 69.724
## 3rd Qu.: 152.288 3rd Qu.: 132.111 3rd Qu.: 129.80 3rd Qu.: 139.332
## Max. : 540.729 Max. : 441.857 Max. : 471.49 Max. : 594.592
## 149 150 151 152
## Min. :-369.19 Min. :-383.887 Min. :-334.753 Min. :-440.09
## 1st Qu.: 5.41 1st Qu.: 5.059 1st Qu.: 8.929 1st Qu.: 10.88
## Median : 71.07 Median : 70.611 Median : 70.477 Median : 71.29
## Mean : 70.86 Mean : 70.405 Mean : 70.591 Mean : 71.00
## 3rd Qu.: 136.03 3rd Qu.: 135.286 3rd Qu.: 131.629 3rd Qu.: 130.52
## Max. : 522.09 Max. : 466.648 Max. : 600.061 Max. : 477.12
## 153 154 155 156
## Min. :-310.78 Min. :-622.179 Min. :-352.68 Min. :-528.009
## 1st Qu.: 10.02 1st Qu.: 4.437 1st Qu.: 14.26 1st Qu.: 2.712
## Median : 71.32 Median : 70.654 Median : 71.07 Median : 70.673
## Mean : 70.89 Mean : 71.201 Mean : 71.22 Mean : 70.373
## 3rd Qu.: 132.26 3rd Qu.: 138.630 3rd Qu.: 128.47 3rd Qu.: 138.312
## Max. : 544.63 Max. : 535.712 Max. : 452.50 Max. : 492.153
## 157 158 159 160
## Min. :-476.60 Min. :-362.375 Min. :-409.067 Min. :-440.09
## 1st Qu.: -14.53 1st Qu.: 8.137 1st Qu.: 2.574 1st Qu.: -15.90
## Median : 71.48 Median : 71.098 Median : 71.297 Median : 71.27
## Mean : 70.70 Mean : 71.126 Mean : 70.835 Mean : 71.03
## 3rd Qu.: 156.64 3rd Qu.: 133.640 3rd Qu.: 140.565 3rd Qu.: 157.85
## Max. : 654.91 Max. : 434.286 Max. : 466.031 Max. : 692.15
## 161 162 163 164
## Min. :-418.75 Min. :-591.47 Min. :-391.97 Min. :-453.848
## 1st Qu.: -10.66 1st Qu.: 9.99 1st Qu.: 14.23 1st Qu.: 4.645
## Median : 70.21 Median : 70.69 Median : 70.73 Median : 70.437
## Mean : 70.52 Mean : 71.58 Mean : 71.14 Mean : 70.295
## 3rd Qu.: 152.93 3rd Qu.: 133.49 3rd Qu.: 128.35 3rd Qu.: 135.923
## Max. : 585.88 Max. : 490.65 Max. : 564.85 Max. : 528.592
## 165 166 167 168
## Min. :-326.618 Min. :-440.536 Min. :-580.106 Min. :-409.545
## 1st Qu.: 6.291 1st Qu.: -6.341 1st Qu.: 8.439 1st Qu.: 4.351
## Median : 69.864 Median : 71.934 Median : 70.540 Median : 69.810
## Mean : 70.480 Mean : 72.102 Mean : 70.612 Mean : 70.366
## 3rd Qu.: 133.691 3rd Qu.: 150.481 3rd Qu.: 133.690 3rd Qu.: 135.866
## Max. : 539.879 Max. : 535.096 Max. : 503.863 Max. : 598.818
## 169 170 171 172
## Min. :-287.425 Min. :-462.577 Min. :-388.90 Min. :-378.3443
## 1st Qu.: 9.419 1st Qu.: -5.674 1st Qu.: -10.71 1st Qu.: -0.7986
## Median : 72.084 Median : 71.299 Median : 71.32 Median : 71.3646
## Mean : 71.174 Mean : 70.888 Mean : 71.20 Mean : 71.2807
## 3rd Qu.: 131.908 3rd Qu.: 147.008 3rd Qu.: 152.54 3rd Qu.: 143.6043
## Max. : 637.971 Max. : 572.258 Max. : 606.39 Max. : 497.3140
## 173 174 175 176
## Min. :-639.527 Min. :-388.873 Min. :-452.220 Min. :-396.80
## 1st Qu.: 2.104 1st Qu.: 8.618 1st Qu.: -7.915 1st Qu.: -11.28
## Median : 71.038 Median : 71.240 Median : 70.309 Median : 70.00
## Mean : 70.525 Mean : 71.196 Mean : 70.430 Mean : 70.42
## 3rd Qu.: 140.635 3rd Qu.: 133.908 3rd Qu.: 148.218 3rd Qu.: 152.42
## Max. : 500.281 Max. : 436.545 Max. : 599.007 Max. : 561.51
## 177 178 179 180
## Min. :-459.70 Min. :-421.020 Min. :-422.059 Min. :-382.77
## 1st Qu.: -11.90 1st Qu.: 1.812 1st Qu.: -3.168 1st Qu.: 13.81
## Median : 72.36 Median : 71.789 Median : 69.887 Median : 70.67
## Mean : 70.89 Mean : 71.333 Mean : 69.974 Mean : 71.26
## 3rd Qu.: 153.48 3rd Qu.: 139.269 3rd Qu.: 144.414 3rd Qu.: 128.31
## Max. : 568.20 Max. : 522.892 Max. : 674.612 Max. : 580.61
## 181 182 183 184
## Min. :-387.553 Min. :-479.822 Min. :-348.754 Min. :-369.070
## 1st Qu.: 1.042 1st Qu.: -9.248 1st Qu.: 1.906 1st Qu.: 3.761
## Median : 69.370 Median : 71.791 Median : 69.888 Median : 70.532
## Mean : 69.878 Mean : 71.676 Mean : 70.416 Mean : 70.628
## 3rd Qu.: 138.233 3rd Qu.: 152.065 3rd Qu.: 138.577 3rd Qu.: 137.312
## Max. : 476.211 Max. : 550.406 Max. : 512.230 Max. : 528.362
## 185 186 187 188
## Min. :-357.042 Min. :-450.936 Min. :-441.62 Min. :-390.45
## 1st Qu.: -4.231 1st Qu.: 9.477 1st Qu.: 12.57 1st Qu.: -10.75
## Median : 71.216 Median : 71.165 Median : 70.39 Median : 69.51
## Mean : 70.567 Mean : 71.091 Mean : 70.53 Mean : 69.50
## 3rd Qu.: 145.872 3rd Qu.: 132.488 3rd Qu.: 128.69 3rd Qu.: 149.93
## Max. : 533.840 Max. : 501.872 Max. : 504.86 Max. : 569.99
## 189 190 191 192
## Min. :-431.133 Min. :-430.2150 Min. :-337.26 Min. :-338.9254
## 1st Qu.: -3.212 1st Qu.: 0.9978 1st Qu.: 15.35 1st Qu.: 0.6339
## Median : 72.297 Median : 71.7018 Median : 71.02 Median : 70.7497
## Mean : 71.826 Mean : 71.0475 Mean : 71.03 Mean : 70.8856
## 3rd Qu.: 147.013 3rd Qu.: 141.6495 3rd Qu.: 125.94 3rd Qu.: 140.7080
## Max. : 645.826 Max. : 485.6389 Max. : 512.04 Max. : 477.4773
## 193 194 195 196
## Min. :-380.459 Min. :-542.354 Min. :-430.491 Min. :-380.21
## 1st Qu.: 5.539 1st Qu.: 2.629 1st Qu.: 0.986 1st Qu.: 12.45
## Median : 70.875 Median : 69.839 Median : 71.057 Median : 71.89
## Mean : 70.774 Mean : 70.341 Mean : 70.131 Mean : 71.81
## 3rd Qu.: 135.294 3rd Qu.: 137.700 3rd Qu.: 139.730 3rd Qu.: 131.73
## Max. : 494.373 Max. : 485.507 Max. : 606.260 Max. : 683.40
## 197 198 199 200
## Min. :-398.364 Min. :-489.07 Min. :-415.836 Min. :-415.38
## 1st Qu.: 6.987 1st Qu.: 10.31 1st Qu.: -2.763 1st Qu.: -9.37
## Median : 71.745 Median : 70.77 Median : 71.198 Median : 70.98
## Mean : 70.722 Mean : 70.60 Mean : 71.530 Mean : 70.90
## 3rd Qu.: 134.627 3rd Qu.: 132.13 3rd Qu.: 146.155 3rd Qu.: 151.15
## Max. : 454.461 Max. : 506.69 Max. : 550.216 Max. : 610.31
## 201 202 203 204
## Min. :-345.56 Min. :-311.910 Min. :-349.414 Min. :-380.86
## 1st Qu.: 10.30 1st Qu.: 7.866 1st Qu.: 8.586 1st Qu.: 11.16
## Median : 70.81 Median : 71.096 Median : 70.635 Median : 71.34
## Mean : 70.50 Mean : 70.674 Mean : 70.496 Mean : 71.12
## 3rd Qu.: 130.95 3rd Qu.: 134.602 3rd Qu.: 131.363 3rd Qu.: 131.23
## Max. : 521.98 Max. : 504.878 Max. : 462.179 Max. : 516.54
## 205 206 207 208
## Min. :-312.517 Min. :-420.042 Min. :-336.308 Min. :-333.82
## 1st Qu.: 8.984 1st Qu.: -8.871 1st Qu.: 4.162 1st Qu.: 12.41
## Median : 71.078 Median : 69.415 Median : 71.359 Median : 70.19
## Mean : 71.254 Mean : 70.771 Mean : 70.674 Mean : 70.53
## 3rd Qu.: 133.021 3rd Qu.: 149.863 3rd Qu.: 137.851 3rd Qu.: 128.67
## Max. : 586.143 Max. : 731.592 Max. : 762.450 Max. : 502.97
## 209 210 211 212
## Min. :-360.04 Min. :-377.0112 Min. :-417.937 Min. :-353.634
## 1st Qu.: 10.57 1st Qu.: -0.1787 1st Qu.: -3.442 1st Qu.: 2.771
## Median : 71.31 Median : 72.2645 Median : 71.552 Median : 69.647
## Mean : 71.18 Mean : 71.6419 Mean : 71.970 Mean : 70.133
## 3rd Qu.: 131.84 3rd Qu.: 142.1273 3rd Qu.: 148.184 3rd Qu.: 137.580
## Max. : 566.88 Max. : 508.0355 Max. : 625.948 Max. : 514.382
## 213 214 215 216
## Min. :-318.183 Min. :-438.9347 Min. :-514.221 Min. :-349.46
## 1st Qu.: 8.408 1st Qu.: 0.2306 1st Qu.: -3.618 1st Qu.: 5.60
## Median : 70.412 Median : 71.5225 Median : 70.246 Median : 71.89
## Mean : 70.665 Mean : 70.2331 Mean : 70.693 Mean : 71.62
## 3rd Qu.: 132.919 3rd Qu.: 141.6554 3rd Qu.: 145.147 3rd Qu.: 137.16
## Max. : 545.482 Max. : 502.0180 Max. : 563.220 Max. : 682.42
## 217 218 219 220
## Min. :-441.01 Min. :-343.90 Min. :-320.14 Min. :-390.376
## 1st Qu.: 11.79 1st Qu.: 5.59 1st Qu.: 14.90 1st Qu.: 8.791
## Median : 71.66 Median : 70.84 Median : 70.36 Median : 71.070
## Mean : 71.46 Mean : 70.21 Mean : 70.40 Mean : 70.862
## 3rd Qu.: 130.61 3rd Qu.: 132.98 3rd Qu.: 125.62 3rd Qu.: 132.334
## Max. : 526.22 Max. : 572.74 Max. : 529.50 Max. : 587.554
## 221 222 223 224
## Min. :-295.135 Min. :-429.82 Min. :-425.352 Min. :-461.33
## 1st Qu.: 6.198 1st Qu.: 7.25 1st Qu.: 9.384 1st Qu.: -13.53
## Median : 70.171 Median : 71.11 Median : 70.837 Median : 70.91
## Mean : 70.188 Mean : 70.92 Mean : 71.104 Mean : 70.42
## 3rd Qu.: 132.930 3rd Qu.: 134.12 3rd Qu.: 131.979 3rd Qu.: 153.79
## Max. : 525.361 Max. : 490.07 Max. : 659.425 Max. : 654.88
## 225 226 227 228
## Min. :-600.3013 Min. :-326.666 Min. :-431.62 Min. :-589.40
## 1st Qu.: 0.5192 1st Qu.: 6.921 1st Qu.: 12.38 1st Qu.: -12.18
## Median : 72.1853 Median : 70.704 Median : 71.80 Median : 70.53
## Mean : 71.6498 Mean : 71.151 Mean : 71.21 Mean : 70.51
## 3rd Qu.: 142.5396 3rd Qu.: 134.602 3rd Qu.: 129.15 3rd Qu.: 153.07
## Max. : 601.9791 Max. : 592.639 Max. : 527.10 Max. : 682.73
## 229 230 231 232
## Min. :-444.10 Min. :-358.282 Min. :-363.49 Min. :-342.939
## 1st Qu.: -7.54 1st Qu.: 3.041 1st Qu.: 7.06 1st Qu.: 7.534
## Median : 71.31 Median : 70.782 Median : 71.50 Median : 68.941
## Mean : 71.21 Mean : 70.903 Mean : 71.23 Mean : 70.605
## 3rd Qu.: 149.35 3rd Qu.: 138.480 3rd Qu.: 134.70 3rd Qu.: 133.818
## Max. : 601.24 Max. : 459.447 Max. : 483.96 Max. : 546.559
## 233 234 235 236
## Min. :-325.42 Min. :-378.60 Min. :-388.874 Min. :-553.34
## 1st Qu.: 11.34 1st Qu.: 13.63 1st Qu.: 2.623 1st Qu.: -35.45
## Median : 70.76 Median : 71.02 Median : 71.271 Median : 69.80
## Mean : 70.98 Mean : 70.67 Mean : 70.837 Mean : 70.90
## 3rd Qu.: 130.75 3rd Qu.: 128.32 3rd Qu.: 138.888 3rd Qu.: 177.82
## Max. : 425.17 Max. : 501.52 Max. : 684.549 Max. : 680.09
## 237 238 239 240
## Min. :-480.42 Min. :-691.550 Min. :-358.38 Min. :-311.200
## 1st Qu.: 11.25 1st Qu.: 3.788 1st Qu.: 15.49 1st Qu.: 8.872
## Median : 70.94 Median : 70.317 Median : 70.82 Median : 70.994
## Mean : 71.28 Mean : 70.278 Mean : 71.45 Mean : 71.142
## 3rd Qu.: 131.42 3rd Qu.: 136.586 3rd Qu.: 127.51 3rd Qu.: 134.622
## Max. : 611.30 Max. : 600.362 Max. : 538.28 Max. : 531.073
## 241 242 243 244
## Min. :-515.76 Min. :-339.104 Min. :-368.27 Min. :-401.390
## 1st Qu.: -12.39 1st Qu.: 6.184 1st Qu.: 13.84 1st Qu.: -2.155
## Median : 70.23 Median : 71.093 Median : 71.66 Median : 70.433
## Mean : 70.88 Mean : 70.911 Mean : 71.59 Mean : 71.209
## 3rd Qu.: 153.91 3rd Qu.: 135.950 3rd Qu.: 130.12 3rd Qu.: 144.105
## Max. : 619.02 Max. : 579.990 Max. : 473.76 Max. : 510.736
## 245 246 247 248
## Min. :-307.45 Min. :-366.48 Min. :-390.630 Min. :-333.72
## 1st Qu.: 12.36 1st Qu.: 14.47 1st Qu.: 7.859 1st Qu.: 10.06
## Median : 71.61 Median : 70.63 Median : 71.872 Median : 71.19
## Mean : 71.64 Mean : 70.66 Mean : 70.670 Mean : 71.80
## 3rd Qu.: 131.54 3rd Qu.: 126.56 3rd Qu.: 133.734 3rd Qu.: 134.58
## Max. : 522.32 Max. : 601.53 Max. : 445.189 Max. : 469.54
## 249 250 251 252
## Min. :-472.951 Min. :-425.085 Min. :-347.91 Min. :-373.68
## 1st Qu.: 6.935 1st Qu.: 4.399 1st Qu.: 11.87 1st Qu.: 10.04
## Median : 71.575 Median : 71.226 Median : 71.14 Median : 70.92
## Mean : 71.310 Mean : 71.161 Mean : 70.75 Mean : 71.05
## 3rd Qu.: 135.068 3rd Qu.: 138.021 3rd Qu.: 129.54 3rd Qu.: 132.33
## Max. : 486.305 Max. : 460.757 Max. : 454.04 Max. : 465.51
## 253 254 255 256
## Min. :-423.881 Min. :-395.948 Min. :-386.174 Min. :-497.783
## 1st Qu.: 8.186 1st Qu.: 9.033 1st Qu.: 2.082 1st Qu.: -3.372
## Median : 70.091 Median : 70.190 Median : 69.719 Median : 72.387
## Mean : 70.607 Mean : 70.216 Mean : 70.422 Mean : 72.121
## 3rd Qu.: 133.012 3rd Qu.: 131.101 3rd Qu.: 138.810 3rd Qu.: 147.858
## Max. : 568.729 Max. : 522.977 Max. : 502.187 Max. : 621.146
## 257 258 259 260
## Min. :-605.819 Min. :-382.822 Min. :-456.737 Min. :-368.745
## 1st Qu.: 2.155 1st Qu.: 7.375 1st Qu.: 9.313 1st Qu.: 9.759
## Median : 69.699 Median : 71.252 Median : 72.507 Median : 71.824
## Mean : 70.235 Mean : 70.706 Mean : 71.858 Mean : 70.984
## 3rd Qu.: 137.394 3rd Qu.: 134.844 3rd Qu.: 134.821 3rd Qu.: 131.864
## Max. : 534.539 Max. : 496.518 Max. : 665.788 Max. : 462.208
## 261 262 263 264
## Min. :-390.215 Min. :-476.163 Min. :-384.22 Min. :-391.73
## 1st Qu.: 6.988 1st Qu.: 5.662 1st Qu.: 12.55 1st Qu.: 13.93
## Median : 71.614 Median : 71.070 Median : 70.99 Median : 70.41
## Mean : 71.166 Mean : 70.572 Mean : 71.25 Mean : 71.05
## 3rd Qu.: 136.552 3rd Qu.: 135.371 3rd Qu.: 130.56 3rd Qu.: 128.53
## Max. : 514.509 Max. : 500.641 Max. : 489.06 Max. : 480.40
## 265 266 267 268
## Min. :-407.087 Min. :-442.34 Min. :-376.880 Min. :-370.37
## 1st Qu.: 9.611 1st Qu.: -5.39 1st Qu.: 8.153 1st Qu.: 10.21
## Median : 70.245 Median : 70.59 Median : 70.230 Median : 71.05
## Mean : 70.525 Mean : 70.88 Mean : 71.123 Mean : 71.47
## 3rd Qu.: 132.181 3rd Qu.: 146.80 3rd Qu.: 134.265 3rd Qu.: 133.23
## Max. : 479.642 Max. : 554.61 Max. : 714.777 Max. : 643.83
## 269 270
## Min. :-418.18 Min. :-371.681
## 1st Qu.: 13.37 1st Qu.: -3.512
## Median : 69.96 Median : 71.508
## Mean : 70.52 Mean : 71.410
## 3rd Qu.: 127.87 3rd Qu.: 146.662
## Max. : 454.41 Max. : 598.613pp_check(model_1)
The pp check function gives the predicted values plots by comparing the observed outcome variable y to the simulated values y rep .In the above graph the simulated values are spread across the graph while the observed value is mostly skewed in the value of 190.
tidy(model_1,conf.int = TRUE,conf.level=0.9999)On the basis of the results, identify a subset of predictors that can help you produce a more concise summary of the data. Then, fit a model with this reduced set of predictors by repeating the steps 1 - 3 above with this reduced subset of predictors.
I used tidy function to identify the subset of predictors that can help to produce the most accurate predictors that are statistically significant with the MAX.HR. As the table suggests that most of the predictors are near to the value zero the null hypothesis can be rejected but there are couple of them that need be considered those are the FBS.over.120,sex ,chest pain type ,ST depression and number of vessels fluro are significant predictor variables.
dataset%>%
select(FBS.over.120,Chest.pain.type,Sex,Max.HR)%>%
ggpairs()
GGpairs helps to find the correlation between the contents present in the dataset1.In the above graph Max.HR is negatively correlated with most of the predictor variables while with the exception with the blood sugar level.
Model-2(Reduced model-1)
I am using ,FBS.over.120,cholesterol and gender as predictors and all the predictor variables are not statistically significant with each other .I am going to set up the heart beat range as the prior_intercept while the prior normal as the week prior normal .Hence it is set as auto scale.
model_2 <- stan_glm(
Max.HR~Age+BP+FBS.over.120+Chest.pain.type+Sex,
data = dataset1, family = gaussian,
prior_intercept = normal(71,65.5),
prior = normal(0, 1, autoscale = TRUE),
prior_aux = exponential(1, autoscale = TRUE),
chains = 4, iter = 5000*2, seed = 84735,prior_PD = TRUE)##
## SAMPLING FOR MODEL 'continuous' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'continuous' NOW (CHAIN 2).
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## Model Info:
## function: stan_glm
## family: gaussian [identity]
## formula: Max.HR ~ Age + BP + FBS.over.120 + Chest.pain.type + Sex
## algorithm: sampling
## sample: 20000 (posterior sample size)
## priors: see help('prior_summary')
## observations: 270
## predictors: 6
##
## Estimates:
## mean sd 10% 50% 90%
## (Intercept) 73.2 244.6 -239.4 74.1 383.7
## Age 0.0 2.6 -3.3 0.0 3.3
## BP 0.0 1.3 -1.7 0.0 1.6
## FBS.over.120 -0.4 65.1 -83.9 -0.4 83.7
## Chest.pain.type 0.2 24.5 -31.3 0.3 31.4
## Sex -0.3 49.3 -63.5 -0.1 63.0
## sigma 23.1 23.2 2.4 16.0 54.1
##
## MCMC diagnostics
## mcse Rhat n_eff
## (Intercept) 1.5 1.0 27451
## Age 0.0 1.0 26700
## BP 0.0 1.0 26469
## FBS.over.120 0.4 1.0 28593
## Chest.pain.type 0.1 1.0 27077
## Sex 0.3 1.0 28108
## sigma 0.1 1.0 28130
## log-posterior 0.0 1.0 8083
##
## For each parameter, mcse is Monte Carlo standard error, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence Rhat=1).The summary of the function determines that the prior intercept is with the mean of 73.2 and the standard deviation 244.6, with the value of 383.7 in 90% confidence interval.The rhat value is 1 for all the predictor variables.Additionally we can find that FBS.over.120 and sex is positively correlated with the heart rate.
mcmc_trace(model_2, size = 0.1)
mcmc_dens_overlay(model_2)
mcmc_acf(model_2)
neff_ratio(model_2)## (Intercept) Age BP FBS.over.120 Chest.pain.type
## 1.37255 1.33500 1.32345 1.42965 1.35385
## Sex sigma
## 1.40540 1.40650rhat(model_2)## (Intercept) Age BP FBS.over.120 Chest.pain.type
## 0.9999251 0.9999163 0.9999555 0.9998619 0.9999468
## Sex sigma
## 0.9998806 0.9998437The mcmc_trance function is used to denote the chains .The chains produced for the above model appers to be mixed well together .Hence they are considered as the stable.
The mcmc overlay function represents the probability density function of each parameter overlap with each other well .In the above graph papers that like sufficiently overlapped with each other.
prior_summary(model_2)## Priors for model 'model_2'
## ------
## Intercept (after predictors centered)
## ~ normal(location = 71, scale = 66)
##
## Coefficients
## Specified prior:
## ~ normal(location = [0,0,0,...], scale = [1,1,1,...])
## Adjusted prior:
## ~ normal(location = [0,0,0,...], scale = [ 2.54, 1.30,65.09,...])
##
## Auxiliary (sigma)
## Specified prior:
## ~ exponential(rate = 1)
## Adjusted prior:
## ~ exponential(rate = 0.043)
## ------
## See help('prior_summary.stanreg') for more detailsThe values we assumed are sufficiently identical to the values produced in the prior summary .Hence the values we considered are mostly correct.
tidy(model_2,conf.int = TRUE,conf.level=0.9999)As the table suggests that most of the predictors are near to the value zero the null hypothesis can be rejected but there are couple of them that need be considered those are the FBS.over.120,sex and chest pain type are significant predictor variables and age,BP is mostly equal to zero.Hence I am dropping them for the next model.Since there is no strong relationship between the BP and Max.HR we can reject the null hypothesis.
pp_check(model_2)
The pp check function gives the predicted values plots by comparing the observed outcome variable y to the simulated values y rep .In the above graph the simulated values are spread across the graph while the observed value is mostly skewed in the value of 190.Additionally we can observe that the simulated values are more skewed than the outcome variables.
Model-3(Reduced model-2)
I am using,FBS.over.120,cholesterol,gender,age,BP,chest.pain.type,number.of.vessels as predictors and considering all the predictor variables are not statistically significant with each other .I am going to set up the heart beat range as the prior_intercept while the prior normal as the week prior normal .Hence it is set as auto scale.
dataset6=dataset%>%select(FBS.over.120,Chest.pain.type,Max.HR)
ggpairs(dataset6)
The explanatory analysis is performed by using the ggpairs function ,where in which the correlation of the heart beat is negative with chest pain type and positive with FBS.over 120.
model_3 <- stan_glm(
Max.HR~FBS.over.120+Chest.pain.type,
data = dataset1, family = gaussian,
prior_intercept = normal(71,65.5),
prior = normal(0, 1, autoscale = TRUE),
prior_aux = exponential(1, autoscale = TRUE),
chains = 4, iter = 5000*2, seed = 84735,prior_PD = TRUE)##
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## Chain 4:summary(model_3)##
## Model Info:
## function: stan_glm
## family: gaussian [identity]
## formula: Max.HR ~ FBS.over.120 + Chest.pain.type
## algorithm: sampling
## sample: 20000 (posterior sample size)
## priors: see help('prior_summary')
## observations: 270
## predictors: 3
##
## Estimates:
## mean sd 10% 50% 90%
## (Intercept) 71.7 102.3 -59.6 71.8 201.3
## FBS.over.120 -0.2 65.0 -84.2 0.5 83.1
## Chest.pain.type -0.1 24.4 -31.2 -0.1 31.3
## sigma 23.0 22.9 2.4 15.8 52.9
##
## MCMC diagnostics
## mcse Rhat n_eff
## (Intercept) 0.8 1.0 16901
## FBS.over.120 0.5 1.0 19042
## Chest.pain.type 0.2 1.0 15501
## sigma 0.2 1.0 21772
## log-posterior 0.0 1.0 8860
##
## For each parameter, mcse is Monte Carlo standard error, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence Rhat=1).The summary of the function determines that the prior intercept is with the mean of 71.7 and the standard deviation 102.3, with the value of 201.3 in 90% confidence interval.The rhat value is 1 for all the predictor variables.Additionally we can find that FBS.over.120,sex,chest pain type and number of vessels of fluro are positively correlated with the heart rate.
mcmc_trace(model_3, size = 0.1)
mcmc_dens_overlay(model_3)
mcmc_acf(model_3)
neff_ratio(model_3)## (Intercept) FBS.over.120 Chest.pain.type sigma
## 0.84505 0.95210 0.77505 1.08860rhat(model_3)## (Intercept) FBS.over.120 Chest.pain.type sigma
## 0.9999069 1.0001365 0.9998953 1.0000749The mcmc_trance function is used to denote the chains .The chains produced for the above model appears to be mixed well together .Hence they are considered as the stable.
The mcmc overlay function represents the probability density function of each parameter overlap with each other well .In the above graph papers that like sufficiently overlapped with each other.
pp_check(model_3)
The pp check function gives the predicted values plots by comparing the observed outcome variable y to the simulated values y rep .In the above graph the simulated values are spread across the graph while the observed value is mostly skewed on the value around 200 .
prior_summary(model_3)## Priors for model 'model_3'
## ------
## Intercept (after predictors centered)
## ~ normal(location = 71, scale = 66)
##
## Coefficients
## Specified prior:
## ~ normal(location = [0,0], scale = [1,1])
## Adjusted prior:
## ~ normal(location = [0,0], scale = [65.09,24.38])
##
## Auxiliary (sigma)
## Specified prior:
## ~ exponential(rate = 1)
## Adjusted prior:
## ~ exponential(rate = 0.043)
## ------
## See help('prior_summary.stanreg') for more detailsThe values that we considered are mostly identical with the observed summary value .
tidy(model_3,conf.int = TRUE,conf.level=0.9999)As the table suggests the two predictor values filtered from the 2nd model is showing the significant relationship with the heart beat .
Model-4(Filtered model)
I am using Chest pain type,FBS over 120,slope of ST and number of vessels of fluro as predictors and considering all the predictor variables are not statistically significant with each other .I am going to set up the heart beat range as the prior_intercept while the prior normal as the week prior normal .Hence it is set as auto scale.
dataset3=dataset%>%select(Chest.pain.type,FBS.over.120,Slope.of.ST,Number.of.vessels.fluro,Max.HR)model_4 <- stan_glm(
Max.HR~Chest.pain.type+FBS.over.120+Slope.of.ST+Number.of.vessels.fluro,
data = dataset3, family = gaussian,
prior_intercept = normal(71,65.5),
prior = normal(0, 1, autoscale = TRUE),
prior_aux = exponential(1, autoscale = TRUE),
chains = 4, iter = 5000*2, seed = 84735,prior_PD = TRUE)##
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## Chain 4:summary(model_4)##
## Model Info:
## function: stan_glm
## family: gaussian [identity]
## formula: Max.HR ~ Chest.pain.type + FBS.over.120 + Slope.of.ST + Number.of.vessels.fluro
## algorithm: sampling
## sample: 20000 (posterior sample size)
## priors: see help('prior_summary')
## observations: 270
## predictors: 5
##
## Estimates:
## mean sd 10% 50% 90%
## (Intercept) 71.6 119.1 -81.6 71.7 224.6
## Chest.pain.type 0.1 24.2 -31.3 0.2 31.3
## FBS.over.120 -0.1 65.9 -84.5 -0.5 84.7
## Slope.of.ST -0.2 37.6 -48.7 -0.3 47.5
## Number.of.vessels.fluro 0.1 24.6 -31.3 0.2 31.5
## sigma 23.1 23.1 2.5 16.0 53.1
##
## MCMC diagnostics
## mcse Rhat n_eff
## (Intercept) 0.8 1.0 21486
## Chest.pain.type 0.2 1.0 22535
## FBS.over.120 0.4 1.0 21497
## Slope.of.ST 0.3 1.0 22650
## Number.of.vessels.fluro 0.2 1.0 20953
## sigma 0.1 1.0 24269
## log-posterior 0.0 1.0 8934
##
## For each parameter, mcse is Monte Carlo standard error, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence Rhat=1).mcmc_trace(model_4, size = 0.1)
mcmc_dens_overlay(model_4)
mcmc_acf(model_4)
neff_ratio(model_4)## (Intercept) Chest.pain.type FBS.over.120
## 1.07430 1.12675 1.07485
## Slope.of.ST Number.of.vessels.fluro sigma
## 1.13250 1.04765 1.21345rhat(model_4)## (Intercept) Chest.pain.type FBS.over.120
## 1.0001173 0.9998513 0.9999540
## Slope.of.ST Number.of.vessels.fluro sigma
## 0.9999339 0.9999719 1.0001102The mcmc_trance function is used to denote the chains .The chains produced for the above model appears to be mixed well together .Hence they are considered as the stable.
The mcmc overlay function represents the probability density function of each parameter overlap with each other well .In the above graph papers that like sufficiently overlapped with each other.
posterior_interval(model_4, prob = 0.90)## 5% 95%
## (Intercept) -123.620426 268.75015
## Chest.pain.type -39.340134 39.69641
## FBS.over.120 -107.124480 108.82824
## Slope.of.ST -62.329850 62.20978
## Number.of.vessels.fluro -40.240539 40.13323
## sigma 1.198026 69.63659pp_check(model_4)
The pp check function gives the predicted values plots by comparing the observed outcome variable y to the simulated values y rep .In the above graph the simulated values are spread across the graph while the observed value is mostly skewed in the value around 190 .
tidy(model_4,conf.int = TRUE,conf.level=0.9999)As the table suggests most of the predictors are statistically significantly correlated with the heart beat .Additionally confidence interval is significantly high towards this predictors.
Compare the full and reduced models using appropriate measures of comparison. Which model would you prefer? Explain by drawing on appropriate evidence.
Lets check the posterior predictor check across all the four models
pp_check(model_1)
pp_check(model_2)
pp_check(model_3)
pp_check(model_4)
The model_1 that is the full model and the model_4 looks almost similar so i am checking by numerical comparison between the both models.
set.seed(2022)
predictions <- posterior_predict(model_1, newdata = dataset)
dim(predictions)## [1] 20000 270set.seed(2022)
predictions_r1 <- posterior_predict(model_2, newdata = dataset)
dim(predictions)## [1] 20000 270set.seed(2022)
predictions_r3 <- posterior_predict(model_3, newdata = dataset)
dim(predictions)## [1] 20000 270set.seed(2022)
predictions_r4 <- posterior_predict(model_4, newdata = dataset)
dim(predictions)## [1] 20000 270ppc_intervals(dataset$Max.HR,
yrep = predictions,
prob = 0.5,
prob_outer = 0.95)
ppc_intervals(dataset$Max.HR,
yrep = predictions_r4,
prob = 0.5,
prob_outer = 0.95)
set.seed(84735)
cv_procedure <- prediction_summary_cv(
model = model_1, data = dataset, k = 10)cv_procedure$foldscv_procedure$cvset.seed(84735)
cv_procedure <- prediction_summary_cv(
model = model_2, data = dataset, k = 10)cv_procedure$foldscv_procedure$cvset.seed(84735)
cv_procedure <- prediction_summary_cv(
model = model_3, data = dataset, k = 10)cv_procedure$foldscv_procedure$cvset.seed(84735)
cv_procedure <- prediction_summary_cv(
model = model_4, data = dataset, k = 10)cv_procedure$foldscv_procedure$cvset.seed(84735)
loo_1 <- loo(model_1)## Warning: Found 270 observations with a pareto_k > 0.7. With this many problematic observations we recommend calling 'kfold' with argument 'K=10' to perform 10-fold cross-validation rather than LOO.loo_2 <- loo(model_2)## Warning: Found 270 observations with a pareto_k > 0.7. With this many problematic observations we recommend calling 'kfold' with argument 'K=10' to perform 10-fold cross-validation rather than LOO.loo_3 <- loo(model_3)## Warning: Found 270 observations with a pareto_k > 0.7. With this many problematic observations we recommend calling 'kfold' with argument 'K=10' to perform 10-fold cross-validation rather than LOO.loo_4 <- loo(model_4)## Warning: Found 270 observations with a pareto_k > 0.7. With this many problematic observations we recommend calling 'kfold' with argument 'K=10' to perform 10-fold cross-validation rather than LOO.loo_compare(loo_1, loo_2, loo_3, loo_4)## elpd_diff se_diff
## model_3 0.000000e+00 0.000000e+00
## model_1 -4.788522e+11 1.842865e+10
## model_4 -1.582883e+12 7.272224e+10
## model_2 -1.770927e+12 3.352085e+10set.seed(84735)
prediction_summary(model = model_1, data = dataset)set.seed(84735)
prediction_summary(model = model_2, data = dataset)set.seed(84735)
prediction_summary(model = model_3, data = dataset)set.seed(84735)
prediction_summary(model = model_4, data = dataset)The model 1 i.e the full model is aggregated with the MAE of 31.1 percent .While the model 2 i.e the desired predicted values only accounted 18.1 percent of the MAE .The reduced third model accounted for 15% MAE and the filtered values present in the fourth model accounted for 20.3% MAE which is highest among all of the models leaving behind the full model .
Conclusions
Address the research questions you presented in the first section by drawing on the results from the previous section.
The problem statement that i am going to mainly address is ’How the data collected by IoMT devices specifically heart rate can be used to predict the diseases in the human body before they mature, and cause severe illnesses,
The questions associated with this problem statement are :
1)How accurate is the prediction of illness in the human body by measuring the fluctuations in the human heart rate?
To answer the first question .By performing linear regression analysis across four different models. I could conclude that by measuring the heart rate solely it is difficult to predict the complications inthe body nevertheless the analysis of the rate in real-time can provide some interesting outcomes. The analysis can give some hints to identify the potential risks, making it easy to avoid and stop the chronic illness before they occur.
What are the most number of heart related diseases can be predicted in the human body can be predicted by analyzing the human heart rate?
As I mentioned earlier human heart rate solely cannot cannot be accountable for predicting the heart complications .But by analyzing the heart rate there will be some useful insight that can be obtained.According to my data set that I considered .There are only limited parameter values that can be analysed by real time human heart beat analysis .The predictor values that can be analysed and having the most statistical significance for the above solution are (Chest.pain.type+FBS.over.120+Slope.of.ST+Number.of.vessels.fluro)
Chest pain type -This is a categorical variable that determines the chest pain category.
FBS over 120-The is a numerical variable where it helps in determining the blood sugar level present in the human body.
The slope of ST -This is the integer variable where it determines the depression state of the human being. Is the user facing any depression or experiencing mental illness?
Several vessels fluro- This is a categorical variable where it has a range from 1 to 4. This variable determines the number of vessels experiencing fat accumulation.
Identify appropriate limitations of your analyses.
There are many limitations of my analysis as my project is related with the medical field .There are several unusual circumstances that can been experienced in the human body .So, the data needs to be transformed based on the analysis with taking the inputs from the doctors and professionals related to medical field in order to suffice the erroneous decisions.The mixed method approach can solve the above problem .It is the combination of qualitative and quantitative data ,where the quantitative data is provided by the health trackers and the qualitative data provided by the medical professionals (or)doctors.
Identify two additional research questions that you can answer by doing a follow-up study that builds on the current study. Focus specifically on whether the same variables suffice or if you additional variables need to be included. What approach for collecting data will be needed.
The additional research questions that I was brainstorming while preforming analysis are :
What are the predictor variables that can be measured, as easy as heart rate and are statistically significant ,that improves the accuracy of the models above?
What else variables (or) diseases can be predicted by analyzing the human heart rate ?
Comment on whether the analyses you produced and the conclusions you have drawn can have any adverse implications for the entities in the population from which the current dataset was sampled. In addition to any speculation you may provide, cite appropriate literature (one or two peer-reviewed publications, as appropriate, are sufficient) to inform your answer.
The dataset i considered is only confined to the small group of people with limited resources .My performed analysis can be true for only some circumstances.Since the medical sciences and techniques involved in measuring the parameters keep on updating in the daily basis .
When coming to transforming my analysis into a larger picture there need more parameters recorded based on the geographical location etc. for accurate analysis. (Hayward, R. A., & Hofer, T. P. (2001))There are many different case studies cited in this paper where it deals with the lack of accurate information is leading reason for the medical errors, and the third most cause of death is the medical errors.
References:
Hayward, R. A., & Hofer, T. P. (2001). Estimating hospital deaths due to medical errors: preventability is in the eye of the reviewer. Jama, 286(4), 415-420.
Reflections
What challenges did you face when conceptualizing your project’s idea in terms of choice of dataset, choice of research questions, and choice of variables?
There are several difficult challenges I addressed in channelizing the idea into the problem statement. The choice of the dataset was extremely hard to fetch with the required parameters but I managed to find one in UCI machinery, which contains a different number of parameters that are unwanted in my research. The variables I filtered out for my analysis are one of the most common problems that the majority of heart-related patients are facing.
The transitioning of the problem statement into research questions was a time-consuming task for me in all of the analyses. The choice of the research questions I formulated by reading this research paper on problems faced and the diagnosis of the problems experienced can be done by measuring the simple parameters(Gunčar, G., Kukar, M., Notar,M., Brvar, M., Černelč, P., Notar, M., & Notar, M. (2018)).
In what specific ways did you address these challenges?
The choice of the research questions I formulated by reading this research paper which mentions, the most common problems faced and the diagnosis of the problems experienced can be done through measuring the simple parameters(Gunčar, G., Kukar, M., Notar, M., Brvar, M., Černelč, P., Notar, M., & Notar, M. (2018)).
Reference:
Gunčar, G., Kukar, M., Notar, M., Brvar, M., Černelč, P., Notar, M., & Notar, M. (2018). An application of machine learning to haematological diagnosis. Scientific reports, 8(1), 1-12.
What did you learn about the Bayesian approach to analyzing data, while working on the project and on the assignments in previous weeks in the course? Identify two-three keys insights you have had as part of your learning process.
There are many insights that I learned in this course which I going to take forward with me for performing analysis. The first insight is creating different kinds of models and measuring their probability. The second insight that was interesting to me is a specification of priors using the Stan_glm function.